Global gauge symmetries and spatial asymptotic boundary conditions in Yang-Mills theory

In Yang-Mills theory on a Euclidean Cauchy surface, the physical gauge group is often taken to be \mathcal{G}^I/\mathcal{G}^∞_0 𝒢 I / 𝒢 0 ∞ , where \mathcal{G}^I 𝒢 I consists of boundary-preserving gauge transformations asymptoting to a constant, and \mathcal{G}^∞_0 𝒢 0 ∞ consists of transformations generated by the Gauss law constraint. We rigorously derive this physical gauge group for both Abelian and non-Abelian theories. A key result is that restricting to \mathcal{G}^I 𝒢 I follows from the structure of the instantaneous state space on which the instantaneous Lagrangian is defined. We extend our analysis to Yang-Mills-Higgs theory, showing that boundary conditions and the physical gauge group differ between the unbroken and broken phases.